Chapter 10 TRƯỜNG ĐIỆN TỪ

Equation 10.9 shows that the wave travels with velocity u in the +z direction.. A negative sign in u>t ± /3z is associated with a wave propagating in the +z di-rection forward traveling

Chapter 10

ELECTROMAGNETIC WAVE PROPAGATION

How far you go in life depends on your being tender with the young, sionate with the aged, sympathetic with the striving, and tolerant of the weak and the strong Because someday in life you will have been all of these.

compas-—GEORGE W CARVER

10.1 INTRODUCTION

Our first application of Maxwell's equations will be in relation to electromagnetic wavepropagation The existence of EM waves, predicted by Maxwell's equations, was first in-vestigated by Heinrich Hertz After several calculations and experiments Hertz succeeded

in generating and detecting radio waves, which are sometimes called Hertzian waves in hishonor

In general, waves are means of transporting energy or information.

Typical examples of EM waves include radio waves, TV signals, radar beams, and lightrays All forms of EM energy share three fundamental characteristics: they all travel athigh velocity; in traveling, they assume the properties of waves; and they radiate outwardfrom a source, without benefit of any discernible physical vehicles The problem of radia-tion will be addressed in Chapter 13

In this chapter, our major goal is to solve Maxwell's equations and derive EM wavemotion in the following media:

1 Free space (<T = 0, s = eo, JX = /x o )

2 Lossless dielectrics (a = 0, e = e,s o , JX = jx r jx o , or a <sC aie)

3 Lossy dielectrics {a # 0, e = E,E O , fx = fx r ix o )

4 Good conductors (a — °°, e = eo, JX = ix r fx o , or a ^S> we)

where w is the angular frequency of the wave Case 3, for lossy dielectrics, is the mostgeneral case and will be considered first Once this general case is solved, we simply

derive other cases (1,2, and 4) from it as special cases by changing the values of a, e, and

ix However, before we consider wave motion in those different media, it is appropriate that

we study the characteristics of waves in general This is important for proper 410

understand-10.2 WAVES IN GENERAL 411

ing of EM waves The reader who is conversant with the concept of waves may skipSection 10.2 Power considerations, reflection, and transmission between two differentmedia will be discussed later in the chapter

10.2 WAVES IN GENERAL

A clear understanding of EM wave propagation depends on a grasp of what waves are ingeneral

A wave is a function of both space and time.

Wave motion occurs when a disturbance at point A, at time t o , is related to what happens at

point B, at time t > t 0 A wave equation, as exemplified by eqs (9.51) and (9.52), is a

partial differential equation of the second order In one dimension, a scalar wave equationtakes the form of

d 2 E 2 d 2 E

r- - U r- = 0

dt 2 dz 2

(10.1)

where u is the wave velocity Equation (10.1) is a special case of eq (9.51) in which the

medium is source free (pv, = 0, J = 0) It can be solved by following procedure, similar tothat in Example 6.5 Its solutions are of the form

(10.2c)

where / and g denote any function of z — ut and z + ut, respectively Examples of such functions include z ± ut, sin k(z ± ut), cos k(z ± ut), and e J k( - z±u '\ where k is a constant It

can easily be shown that these functions all satisfy eq (10.1)

If we particularly assume harmonic (or sinusoidal) time dependence e J0 ", eq (10.1)

becomes

d 2 E,

where /3 = u/u and E s is the phasor form of E The solution to eq (10.3) is similar to

Case 3 of Example 6.5 [see eq (6.5.12)] With the time factor inserted, the possible tions to eq (10.3) are

(10.4b)

412 B Electromagnetic Wave Propagation

and

= Ae i{<M " 0z) + Be j(ut+fiz) (10.4c)

where A and B are real constants.

For the moment, let us consider the solution in eq (10.4a) Taking the imaginary part

of this equation, we have

2 A is called the amplitude of the wave and has the same units as E.

3 (ox - /3z) is the phase (in radians) of the wave; it depends on time t and space able z.

vari-4 w is the angular frequency (in radians/second); 0 is the phase constant or wave

number (in radians/meter).

Due to the variation of E with both time t and space variable z, we may plot £ as a function of t by keeping z constant and vice versa The plots of E(z, t = constant) and

E(t, z = constant) are shown in Figure 10.1(a) and (b), respectively From Figure 10.1(a),

we observe that the wave takes distance X to repeat itself and hence X is called the

wave-length (in meters) From Figure 10.1(b), the wave takes time T to repeat itself;

conse-quently T is known as the period (in seconds) Since it takes time T for the wave to travel distance X at the speed u, we expect

(10.7b)and

/

(10.7c)

3X\

2 \

/ 2 X /

• A sin(co/ - &z): (a) with constant t,

we expect from eqs (10.6) and (10.7) that

(10.8)

Equation (10.8) shows that for every wavelength of distance traveled, a wave undergoes aphase change of 2TT radians

We will now show that the wave represented by eq (10.5) is traveling with a velocity

u in the +z direction To do this, we consider a fixed point P on the wave We sketch

eq (10.5) at times t = 0, 774, and 772 as in Figure 10.2 From the figure, it is evident that

as the wave advances with time, point P moves along +z direction Point P is a point of

constant phase, therefore

ut - j3z = constant

or

dz

(10.9)

414 Electromagnetic Wave Propagation

Figure 10.2 Plot of E(z, t) = A

sin(cot - /3z) at time (a) t = 0, (b)

t = T/4, (c) t = 772; P moves along +z direction with velocity u.

(c) t = Tj2

which is the same as eq (10.7b) Equation (10.9) shows that the wave travels with velocity

u in the +z direction Similarly, it can be shown that the wave B sin (cof + (5z) in

eq (10.4b) is traveling with velocity u in the — z direction.

In summary, we note the following:

1 A wave is a function of both time and space

2 Though time / = 0 is arbitrarily selected as a reference for the wave, a wave iswithout beginning or end

3 A negative sign in (u>t ± /3z) is associated with a wave propagating in the +z

di-rection (forward traveling or positive-going wave) whereas a positive sign

indi-cates that a wave is traveling in the —z direction (backward traveling or

negative-going wave)

4 Since sin (~\p) = -sin ^ = sin (\j/ ± ir), whereas cos(-i/<) = cos \p,

sin (\j/ ± itl2) = ± cos \[/

sin (\p ± ir) = —sin \j/

cos (\p ± if 12) = + sin \p cos (\j/ ± IT) = —cos \f/

(10.10a)(10.10b)(10.10c)(lO.lOd)

where \p = u>t ± ffz- With eq (10.10), any time-harmonic wave can be represented

in the form of sine or cosine

10.2 WAVES IN GENERAL 415

TABLE 10.1 Electromagnetic Spectrum

EM Phenomena Examples of Uses Approximate Frequency Range Cosmic rays

Gamma rays X-rays Ultraviolet radiation Visible light Infrared radiation Microwave waves Radio waves

Physics, astronomy Cancer therapy X-ray examination Sterilization Human vision Photography Radar, microwave relays, satellite communication UHF television

VHF television, FM radio Short-wave radio

AM radio

10 14 GHz and above 10'°-10 13 GHz

10 8 -10 9 GHz

10 6 -10 8 GHz

10 5 -10 6 GHz

10 3 -10 4 GHz 3-300 GHz 470-806 MHz 54-216 MHz 3-26 MHz 535-1605 kHz

A large number of frequencies visualized in numerical order constitute a spectrum.

Table 10.1 shows at what frequencies various types of energy in the EM spectrum occur.Frequencies usable for radio communication occur near the lower end of the EM spectrum

As frequency increases, the manifestation of EM energy becomes dangerous to humanbeings.1 Microwave ovens, for example, can pose a hazard if not properly shielded Thepractical difficulties of using EM energy for communication purposes also increase as fre-quency increases, until finally it can no longer be used As communication methodsimprove, the limit to usable frequency has been pushed higher Today communicationsatellites use frequencies near 14 GHz This is still far below light frequencies, but in theenclosed environment of fiber optics, light itself can be used for radio communication.2

EXAMPLE 10.1 The electric field in free space is given by

E = 50 cos (108r + &x) a y V/m(a) Find the direction of wave propagation

(b) Calculate /3 and the time it takes to travel a distance of A/2

(c) Sketch the wave at t = 0, 774, and 772.

Solution:

(a) From the positive sign in (tot + /3x), we infer that the wave is propagating along

This will be confirmed in part (c) of this example

'See March 1987 special issue of IEEE Engineering in Medicine and Biology Magazine on "Effects

of EM Radiation."

2See October 1980 issue of IEEE Proceedings on "Optical-Fiber Communications."

416 • Electromagnetic Wave Propagation

(b) In free space, u = c.

c 3 X 10s

or

/3 = 0.3333 rad/m

If 7 is the period of the wave, it takes 7 seconds to travel a distance X at speed c Hence to

travel a distance of X/2 will take

as obtained before

(c) At t = O,E y = 50 cos I3x

At t = 7/4, E y = 50 cos (co • — + /3JC I = 50 cos (fix + TT/2)

10.3 WAVE PROPAGATION IN LOSSY DIELECTRICS 417

(c) Sketch the wave at time t x

Answer: (a) 0.667 rad/m, 9.425 m, 31.42 ns, (b) 3.927 ns, (c) see Figure 10.4.

0.3 WAVE PROPAGATION IN LOSSY DIELECTRICS

As mentioned in Section 10.1, wave propagation in lossy dielectrics is a general case fromwhich wave propagation in other types of media can be derived as special cases Therefore,this section is foundational to the next three sections

418 • Electromagnetic Wave Propagation

0 1 " >

Figure 10.4 For Practice Exercise 10.1(c).

A lossy dielectric is a medium in which an EM wave loses power as it propagates

due to poor conduction

In other words, a lossy dielectric is a partially conducting medium (imperfect dielectric or

imperfect conductor) with a ¥= 0, as distinct from a lossless dielectric (perfect or good electric) in which a = 0.

di-Consider a linear, isotropic, homogeneous, lossy dielectric medium that is charge free

(p v = 0) Assuming and suppressing the time factor e j "', Maxwell's equations (see Table

10.3 WAVE PROPAGATION IN LOSSY DIELECTRICS 419

and y is called the propagation constant (in per meter) of the medium By a similar

proce-dure, it can be shown that for the H field,

V2HS - y 2 K s = 0 (10.19)

Equations (10.17) and (10.19) are known as homogeneous vector Helmholtz 's equations or simply vector wave equations In Cartesian coordinates, eq (10.17), for example, is equiv-

alent to three scalar wave equations, one for each component of E along ax, a y , and az

Since y in eqs (10.17) to (10.19) is a complex quantity, we may let

Without loss of generality, if we assume that the wave propagates along +az and that

Es has only an x-component, then

420 B Electromagnetic Wave Propagation

This is a scalar wave equation, a linear homogeneous differential equation, with solution(see Case 2 in Example 6.5)

EJx) = E o e' yz + E' o e yz (10.28)

where E o and E' o are constants The fact that the field must be finite at infinity requires that

E' o = 0 Alternatively, because e iz denotes a wave traveling along —az whereas we assumewave propagation along az, E' o = 0 Whichever way we look at it, E' o = 0 Inserting the

time factor e jo " into eq (10.28) and using eq (10.20), we obtain

Efc t) = Re a J = Re (E o e- az e ji "'- 0z) a x )

or

Efo i) = E o e~ az cos(at - j3z)a x (10.29)

A sketch of |E| at times t = 0 and t = At is portrayed in Figure 10.5, where it is evident

that E has only an x-component and it is traveling along the +z-direction Having obtained

E(z, t), we obtain H(z, t) either by taking similar steps to solve eq (10.19) or by using eq.

(10.29) in conjunction with Maxwell's equations as we did in Example 9.8 We will tually arrive at

even-H(z, t) = Re (H o e-ayM-ft) ) (10.30)where

and 77 is a complex quantity known as the intrinsic impedance (in ohms) of the medium It

can be shown by following the steps taken in Example 9.8 that

Notice from eqs (10.29) and (10.34) that as the wave propagates along az, it decreases or

attenuates in amplitude by a factor e~ az , and hence a is known as the attenuation constant

or attenuation factor of the medium It is a measure of the spatial rate of decay of the wave

in the medium, measured in nepers per meter (Np/m) or in decibels per meter (dB/m) An

attenuation of 1 neper denotes a reduction to e~ l of the original value whereas an increase

of 1 neper indicates an increase by a factor of e Hence, for voltages

leads H (or H lags E) by 6 V Finally, we notice that the ratio of the magnitude of the

con-duction current density J to that of the displacement current density J d in a lossy medium

422 Electromagnetic Wave Propagation

where tan 6 is known as the loss tangent and d is the loss angle of the medium as illustrated

in Figure 10.6 Although a line of demarcation between good conductors and lossy

di-electrics is not easy to make, tan 6 or 6 may be used to determine how lossy a medium is.

A medium is said to be a good (lossless or perfect) dielectric if tan d is very small (<j <SC we) or a good conductor if tan 0 is very large (a ^5> we) From the viewpoint of

wave propagation, the characteristic behavior of a medium depends not only on its

consti-tutive parameters a, e, and fx but also on the frequency of operation A medium that is

re-garded as a good conductor at low frequencies may be a good dielectric at high cies Note from eqs (10.33) and (10.37) that

and e' = e, s" = a/w; s c is called the complex permittivity of the medium We observe that

the ratio of e" to e' is the loss tangent of the medium; that is,

e a tan d = — = —

In subsequent sections, we will consider wave propagation in other types of media,which may be regarded as special cases of what we have considered here Thus we willsimply deduce the governing formulas from those obtained for the general case treated inthis section The student is advised not just to memorize the formulas but to observe howthey are easily obtained from the formulas for the general case

J ds = Figure 10.6 Loss angle of a lossy medium.

J

10.4 PLANE WAVES IN LOSSLESS DIELECTRICS

In a lossless dielectric, a <$C we It is a special case of that in Section 10.3 except that

i 0.5 PLANE WAVES IN FREE SPACE

This is a special case of what we considered in Section 10.3 In this case,

This may also be regarded as a special case of Section 10.4 Thus we simply replace e by

eo and \k by /xo in eq (10.43) or we substitute eq (10.45) directly into eqs (10.23) and(10.24) Either way, we obtain

424 • Electromagnetic Wave Propagation

By substituting the constitutive parameters in eq (10.45) into eq (10.33), d v = 0 and

V = ^oi where rjo is called the intrinsic impedance of free space and is given by

a k X a£ = a H

or

X aH =

-Figure 10.7 (a) Plot of E and H as

func-tions of z at t = 0; (b) plot of E and H at

z = 0 The arrows indicate instantaneous

values.

(a)

-E = E o cos oj/ a x

H = H o cos ut a y

10.6 PLANE WAVES IN G O O D CONDUCTORS 425or

Both E and H fields (or EM waves) are everywhere normal to the direction of wave

prop-agation, a k That means, the fields lie in a plane that is transverse or orthogonal to the

di-rection of wave propagation They form an EM wave that has no electric or magnetic field

components along the direction of propagation; such a wave is called a transverse

electro-magnetic (TEM) wave Each of E and H is called a uniform plane wave because E (or H)

has the same magnitude throughout any transverse plane, defined by z = constant The rection in which the electric field points is the polarization of a TEM wave.3 The wave in

di-eq (10.29), for example, is polarized in the ^-direction This should be observed in Figure10.7(b), where an illustration of uniform plane waves is given A uniform plane wavecannot exist physically because it stretches to infinity and would represent an infiniteenergy However, such waves are characteristically simple but fundamentally important.They serve as approximations to practical waves, such as from a radio antenna, at distancessufficiently far from radiating sources Although our discussion after eq (10.48) deals withfree space, it also applies for any other isotropic medium

0.6 PLANE WAVES IN GOOD CONDUCTORS

This is another special case of that considered in Section 10.3 A perfect, or good

conduc-tor, is one in which a ^S> we so that a/we —> °o; that is,

426 P Electromagnetic Wave Propagation

then

H = az cos(co? — &z — 45°) a (10.53b)

Therefore, as E (or H) wave travels in a conducting medium, its amplitude is attenuated by

the factor e~ az The distance <5, shown in Figure 10.8, through which the wave amplitude

decreases by a factor e~ l (about 37%) is called skin depth or penetration depth of the

medium; that is,

Figure 10.8 Illustration of skin depth

10.6 PLANE WAVES IN G O O D CONDUCTORS 427 TABLE 10.2 Skin

Frequency (Hz)

Skin depth (mm)

Depth in

10 60 20.8 8.6

*For copper, a = 5.8 X IO7 mhos/m, fi = ft,,, <5 = 66.1/ vf (in mm).

Also for good conductors, eq (10.53a) can be written as

E = E a e~ dh cos o>t )a x

showing that 5 measures the exponential damping of the wave as it travels through the ductor The skin depth in copper at various frequencies is shown in Table 10.2 From thetable, we notice that the skin depth decreases with increase in frequency Thus, E and Hcan hardly propagate through good conductors

con-The phenomenon whereby field intensity in a conductor rapidly decreases is known as

skin effect The fields and associated currents are confined to a very thin layer (the skin) of

the conductor surface For a wire of radius a, for example, it is a good approximation at

high frequencies to assume that all of the current flows in the circular ring of thickness 5 asshown in Figure 10.9 Skin effect appears in different guises in such problems as attenua-tion in waveguides, effective or ac resistance of transmission lines, and electromagneticshielding It is used to advantage in many applications For example, because the skindepth in silver is very small, the difference in performance between a pure silver compo-nent and a silver-plated brass component is negligible, so silver plating is often used toreduce material cost of waveguide components For the same reason, hollow tubular con-ductors are used instead of solid conductors in outdoor television antennas Effective elec-tromagnetic shielding of electrical devices can be provided by conductive enclosures a fewskin depths in thickness

The skin depth is useful in calculating the ac resistance due to skin effect The tance in eq (5.16) is called the dc resistance, that is,

Figure 10.9 Skin depth at high frequencies, 5 <SC a.

428 Electromagnetic Wave Propagation

We define the surface or skin resistance R s (in fl/m2) as the real part of the 77 for a goodconductor Thus from eq (10.55)

(10.56)

This is the resistance of a unit width and unit length of the conductor It is equivalent to the

dc resistance for a unit length of the conductor having cross-sectional area 1 X 5 Thus for

a given width w and length €, the ac resistance is calculated using the familiar dc resistancerelation of eq (5.16) and assuming a uniform current flow in the conductor of thickness 6,that is,

Since 6 <3C a at high frequencies, this shows that /?ac is far greater than R dc In general, the

ratio of the ac to the dc resistance starts at 1.0 for dc and very low frequencies and creases as the frequency increases Also, although the bulk of the current is nonuniformlydistributed over a thickness of 56 of the conductor, the power loss is the same as though itwere uniformly distributed over a thickness of 6 and zero elsewhere This is one morereason why 5 is referred to as the skin depth

in-EXAMPLE 10.2 A lossy dielectric has an intrinsic impedance of 200 /30° fi at a particular frequency If, at

that frequency, the plane wave propagating through the dielectric has the magnetic fieldcomponent

10.6 PLANE WAVES IN GOOD CONDUCTORS B 429

AlsoWo = 10, so

H, - = 77 = 200 rW = 200 e J * 16 -> E o = 2000e"r/6

Except for the amplitude and phase difference, E and H always have the same form Hence

E = Re (2000e; 7 rV7V"'a£)or

m

5 = - = 2 V 3 = 3.4641a

The wave has an E z component; hence it is polarized along the z-direction

430 Electromagnetic Wave Propagation

PRACTICE EXERCISE 10.2

A plane wave propagating through a medium with e r — 8, ix r - 2 has E = 0.5

e~^3 sin(108f - @z) a x V/m Determine(a) 0

(b) The loss tangent(c) Wave impedance(d) Wave velocity(e) H field

Answer: (a) 1.374 rad/m, (b) 0.5154, (c) 177.72 /13.63° 0, (d) 7.278 X 107 m/s,

(e) 2.%\le~ M sin(1081 - 0z - 13.63°)ay mA/m.

EXAMPLE 10.3 In a lossless medium for which -q = 60ir, ix r = 1, and H = —0.1 cos (cof — z) ax +

0.5 sin (cor — z)& y A/m, calculate e r, co, and E.

Method 1: To use the techniques developed in this chapter, we let

E = H, + H 2

10.6 PLANE WAVES IN G O O D CONDUCTORS 431where Hj = -0.1 cos (uf - z) ax and H2 = 0.5 sin (wt - z) a y and the correspondingelectric field

E = E, + E7

where Ej = E lo cos (cof - z) a£i and E2 = E 2o sin (cof - z) aEi Notice that although H

has components along a x and ay, it has no component along the direction of propagation; it

is therefore a TEM wave

= H 2o cos {bit - z) ax + H lo sin (wf - z)ay

where H lo = - 0 1 and//2 o = 0.5 Hence

432 8 Electromagnetic Wave Propagation

A plane wave traveling in the +)>-direction in a lossy medium (e r = 4, \x r = 1,

cr = 10"2 mhos/m) has E = 30 cos (109?r t + x/4) az V/m at y = 0 Find

(a) E at y = 1 m, / = 2 ns

(b) The distance traveled by the wave to have a phase shift of 10°

(c) The distance traveled by the wave to have its amplitude reduced by 40%

(d) H at y = 2 m, t = 2 ns

Answer: (a) 2.787az V/m, (b) 8.325 mm, (c) 542 mm, (d) -4.71a, mA/m

XAMPLE10.5 A plane wave E = E o cos (u>t - j3z) ax is incident on a good conductor at z = 0 Find the

current density in the conductor

434 Electromagnetic Wave Propagation

The constant B must be zero because J sx is finite as z~> °° But in a good conductor,

Due to the current density of Example 10.5, find the magnitude of the total current

through a strip of the conductor of infinite depth along z and width w along y.

Answer:

V~2

EXAMPLE 10.6 For the copper coaxial cable of Figure 7.12, let a = 2 mm, b = 6 mm, and t = 1 mm

Cal-culate the resistance of 2 m length of the cable at dc and at 100 MHz

10.7 POWER AND THE POYNTING VECTOR 435

A t / = 100 MHz,

Rsl _ I

w o82ira 2-KO V o 2

Hence,

R ac = 0.41 + 0.1384 = 0.5484 U

which is about 150 times greater than R dc Thus, for the same effective current i, the ohmic

loss (i 2 R) of the cable at 100 MHz is far greater than the dc power loss by a factor of 150.

0.7 POWER AND THE POYNTING VECTOR

As mentioned before, energy can be transported from one point (where a transmitter islocated) to another point (with a receiver) by means of EM waves The rate of such energytransportation can be obtained from Maxwell's equations:

V X E = -J

dt dE

—

dt

(10.58a)

(10.58b)

436 (ft Electromagnetic Wave Propagation

Dotting both sides of eq (10.58b) with E gives

From eq (10.58a),

and thus eq (10.60) becomes

the medium is conducting (a # 0) The quantity E X H on the left-hand side of eq (10.63)

is known as the Poynting vector SP in watts per square meter (W/m2); that is,

4After J H Poynting, "On the transfer of energy in the electromagnetic field," Phil Trans., vol 174,

1883, p 343.

10.7 POWER AND THE POYNTINC VECTOR 437

It represents the instantaneous power density vector associated with the EM field at a givenpoint The integration of the Poynting vector over any closed surface gives the net powerflowing out of that surface

Poynting's theorem stales th;it the nel power flowing out of a given volume i i\

equal to the lime rate of decrease in the energy stored wilhin r minus the conductionlosses

The theorem is illustrated in Figure, 10.10

It should be noted that 9s is normal to both E and H and is therefore along the

direc-tion of wave propagadirec-tion a k for uniform plane waves Thus

438 Electromagnetic Wave Propagation

since cos A cos B = — [cos (A — 5) + cos (A + B)] To determine the time-average

Poynting vector 2?ave(z) (in W/m2), which is of more practical value than the instantaneous

Poynting vector 2P(z, t), we integrate eq (10.66) over the period T = 2ir/u>; that is,

(a) e r , r;

(b) The time-average power carried by the wave

(c) The total power crossing 100 cm 2 of plane 2x + y = 5

440 • Electromagnetic Wave Propagation

PRACTICE EXERCISE 10.7

In free space, H = 0.2 cos (uit — /3x) a z A/m Find the total power passing through:

(a) A square plate of side 10 cm on plane x + z = 1 (b) A circular disc of radius 5 cm on plane x = 1.

mitted depends on the constitutive parameters (e, ju, a) of the two media involved Here we

will assume that the incident wave plane is normal to the boundary between the media;oblique incidence of plane waves will be covered in the next section after we understandthe simpler case of normal incidence

Suppose that a plane wave propagating along the +z-direction is incident normally on

the boundary z = 0 between medium 1 (z < 0) characterized by er,, e u fi x and medium

2 (z > 0) characterized by a 2 , e2, /*2> as shown in Figure 10.11 In the figure, subscripts /,

r, and t denote incident, reflected, and transmitted waves, respectively The incident,

re-flected, and transmitted waves shown in Figure 10.11 are obtained as follows:

Incident Wave:

(E,, H,) is traveling along +az in medium 1 If we suppress the time factor e Jo " and assume

that

E ls (z) = E io e- y ' z ax (10.71)then

H,,(z) = H io e-" z a, = ^ e~™ av (10.72)

Reflected Wave:

(E n Hr) is traveling along -& z in medium 1 If